Clustering and classification (Exercise 4)

library(MASS);library(tidyverse);library(corrplot)
## -- Attaching packages ------------------------------------------------------------------------------------- tidyverse 1.2.1 --
## v ggplot2 3.2.1     v purrr   0.3.3
## v tibble  2.1.3     v dplyr   0.8.3
## v tidyr   1.0.0     v stringr 1.4.0
## v readr   1.3.1     v forcats 0.4.0
## -- Conflicts ---------------------------------------------------------------------------------------- tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag()    masks stats::lag()
## x dplyr::select() masks MASS::select()
## corrplot 0.84 loaded
data("Boston")

The data has 506 observations and 14 variables. Data is about housing values in suburbs of Boston.

Explort dataset

str(Boston)
## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
summary(Boston)
##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08204   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00
pairs(Boston)

Correlate all variables

cor_matrix<-cor(Boston)
cor_matrix%>%round(2)
##          crim    zn indus  chas   nox    rm   age   dis   rad   tax
## crim     1.00 -0.20  0.41 -0.06  0.42 -0.22  0.35 -0.38  0.63  0.58
## zn      -0.20  1.00 -0.53 -0.04 -0.52  0.31 -0.57  0.66 -0.31 -0.31
## indus    0.41 -0.53  1.00  0.06  0.76 -0.39  0.64 -0.71  0.60  0.72
## chas    -0.06 -0.04  0.06  1.00  0.09  0.09  0.09 -0.10 -0.01 -0.04
## nox      0.42 -0.52  0.76  0.09  1.00 -0.30  0.73 -0.77  0.61  0.67
## rm      -0.22  0.31 -0.39  0.09 -0.30  1.00 -0.24  0.21 -0.21 -0.29
## age      0.35 -0.57  0.64  0.09  0.73 -0.24  1.00 -0.75  0.46  0.51
## dis     -0.38  0.66 -0.71 -0.10 -0.77  0.21 -0.75  1.00 -0.49 -0.53
## rad      0.63 -0.31  0.60 -0.01  0.61 -0.21  0.46 -0.49  1.00  0.91
## tax      0.58 -0.31  0.72 -0.04  0.67 -0.29  0.51 -0.53  0.91  1.00
## ptratio  0.29 -0.39  0.38 -0.12  0.19 -0.36  0.26 -0.23  0.46  0.46
## black   -0.39  0.18 -0.36  0.05 -0.38  0.13 -0.27  0.29 -0.44 -0.44
## lstat    0.46 -0.41  0.60 -0.05  0.59 -0.61  0.60 -0.50  0.49  0.54
## medv    -0.39  0.36 -0.48  0.18 -0.43  0.70 -0.38  0.25 -0.38 -0.47
##         ptratio black lstat  medv
## crim       0.29 -0.39  0.46 -0.39
## zn        -0.39  0.18 -0.41  0.36
## indus      0.38 -0.36  0.60 -0.48
## chas      -0.12  0.05 -0.05  0.18
## nox        0.19 -0.38  0.59 -0.43
## rm        -0.36  0.13 -0.61  0.70
## age        0.26 -0.27  0.60 -0.38
## dis       -0.23  0.29 -0.50  0.25
## rad        0.46 -0.44  0.49 -0.38
## tax        0.46 -0.44  0.54 -0.47
## ptratio    1.00 -0.18  0.37 -0.51
## black     -0.18  1.00 -0.37  0.33
## lstat      0.37 -0.37  1.00 -0.74
## medv      -0.51  0.33 -0.74  1.00
corrplot(cor_matrix, method="circle", type="upper", cl.pos="b", tl.pos="d", tl.cex=0.6)

There is strongly positive correlation bewtween “rad” and “tax” and stongly negative correlation between “age” and “dis” and between “lstat” and “medv”.

Standardized data

boston_scaled<-scale(Boston)
summary(boston_scaled)
##       crim                 zn               indus        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202  
##       chas              nox                rm               age         
##  Min.   :-0.2723   Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331  
##  1st Qu.:-0.2723   1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366  
##  Median :-0.2723   Median :-0.1441   Median :-0.1084   Median : 0.3171  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.:-0.2723   3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059  
##  Max.   : 3.6648   Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164  
##       dis               rad               tax             ptratio       
##  Min.   :-1.2658   Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047  
##  1st Qu.:-0.8049   1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876  
##  Median :-0.2790   Median :-0.5225   Median :-0.4642   Median : 0.2746  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6617   3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058  
##  Max.   : 3.9566   Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372  
##      black             lstat              medv        
##  Min.   :-3.9033   Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.: 0.2049   1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median : 0.3808   Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.4332   3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 0.4406   Max.   : 3.5453   Max.   : 2.9865

Covert matrix data to dafaframe

class(boston_scaled)
## [1] "matrix"
boston_scaled<-as.data.frame(boston_scaled)

Create a catergorical variable with the variable “Crime”

summary("crim")
##    Length     Class      Mode 
##         1 character character
bins<-quantile(boston_scaled$crim)
bins
##           0%          25%          50%          75%         100% 
## -0.419366929 -0.410563278 -0.390280295  0.007389247  9.924109610
crime<-cut(boston_scaled$crim, breaks=bins, include.lowest=TRUE, label=c("low","med_low","med_high", "high"))
crime
##   [1] low      low      low      low      low      low      med_low 
##   [8] med_low  med_low  med_low  med_low  med_low  med_low  med_high
##  [15] med_high med_high med_high med_high med_high med_high med_high
##  [22] med_high med_high med_high med_high med_high med_high med_high
##  [29] med_high med_high med_high med_high med_high med_high med_high
##  [36] low      med_low  low      med_low  low      low      med_low 
##  [43] med_low  med_low  med_low  med_low  med_low  med_low  med_low 
##  [50] med_low  med_low  low      low      low      low      low     
##  [57] low      low      med_low  med_low  med_low  med_low  med_low 
##  [64] med_low  low      low      low      low      med_low  med_low 
##  [71] med_low  med_low  med_low  med_low  low      med_low  med_low 
##  [78] med_low  low      med_low  low      low      low      low     
##  [85] low      low      low      low      low      low      low     
##  [92] low      low      low      low      med_low  med_low  med_low 
##  [99] low      low      med_low  med_low  med_low  med_low  med_low 
## [106] med_low  med_low  med_low  med_low  med_high med_low  med_low 
## [113] med_low  med_low  med_low  med_low  med_low  med_low  med_low 
## [120] med_low  low      low      med_low  med_low  med_low  med_low 
## [127] med_high med_high med_high med_high med_high med_high med_high
## [134] med_high med_high med_high med_high med_high med_low  med_high
## [141] med_high med_high med_high high     med_high med_high med_high
## [148] med_high med_high med_high med_high med_high med_high med_high
## [155] med_high med_high med_high med_high med_high med_high med_high
## [162] med_high med_high med_high med_high med_high med_high med_high
## [169] med_high med_high med_high med_high med_low  med_low  med_low 
## [176] low      low      low      low      low      low      low     
## [183] med_low  med_low  med_low  low      low      low      med_low 
## [190] med_low  med_low  low      med_low  low      low      low     
## [197] low      low      low      low      low      low      low     
## [204] low      low      med_low  med_low  med_low  med_low  med_high
## [211] med_low  med_high med_low  med_low  med_high med_low  low     
## [218] low      med_low  med_low  med_high med_high med_high med_high
## [225] med_high med_high med_high med_high med_high med_high med_high
## [232] med_high med_high med_high med_high med_high med_high med_high
## [239] med_low  med_low  med_low  med_low  med_low  med_low  med_low 
## [246] med_low  med_high med_low  med_low  med_low  med_low  med_low 
## [253] med_low  med_high low      low      low      med_high med_high
## [260] med_high med_high med_high med_high med_high med_high med_high
## [267] med_high med_high med_high med_low  med_high med_low  med_low 
## [274] med_low  low      med_low  med_low  low      low      med_low 
## [281] low      low      low      low      low      low      low     
## [288] low      low      low      low      low      low      med_low 
## [295] low      med_low  low      med_low  low      low      low     
## [302] low      med_low  med_low  low      low      low      low     
## [309] med_high med_high med_high med_high med_high med_high med_high
## [316] med_low  med_high med_low  med_high med_high med_low  med_low 
## [323] med_high med_high med_high med_low  med_high med_low  low     
## [330] low      low      low      low      low      low      low     
## [337] low      low      low      low      low      low      low     
## [344] low      low      low      low      low      low      low     
## [351] low      low      low      low      low      med_low  high    
## [358] high     high     high     high     high     high     high    
## [365] med_high high     high     high     high     high     high    
## [372] high     high     high     high     high     high     high    
## [379] high     high     high     high     high     high     high    
## [386] high     high     high     high     high     high     high    
## [393] high     high     high     high     high     high     high    
## [400] high     high     high     high     high     high     high    
## [407] high     high     high     high     high     high     high    
## [414] high     high     high     high     high     high     high    
## [421] high     high     high     high     high     high     high    
## [428] high     high     high     high     high     high     high    
## [435] high     high     high     high     high     high     high    
## [442] high     high     high     high     high     high     high    
## [449] high     high     high     high     high     high     high    
## [456] high     high     high     high     high     high     high    
## [463] high     high     high     med_high high     high     high    
## [470] high     high     high     med_high high     high     high    
## [477] high     high     high     high     high     high     high    
## [484] med_high med_high med_high high     high     med_low  med_low 
## [491] med_low  med_low  med_low  med_low  med_high med_low  med_high
## [498] med_high med_low  med_low  med_low  low      low      low     
## [505] med_low  low     
## Levels: low med_low med_high high
table(crime)
## crime
##      low  med_low med_high     high 
##      127      126      126      127

Remove original “crim” from the dataset and add the new categorial variable

boston_scaled<-dplyr::select(boston_scaled,-crim)
boston_scaled
##              zn       indus       chas         nox           rm
## 1    0.28454827 -1.28663623 -0.2723291 -0.14407485  0.413262920
## 2   -0.48724019 -0.59279438 -0.2723291 -0.73953036  0.194082387
## 3   -0.48724019 -0.59279438 -0.2723291 -0.73953036  1.281445551
## 4   -0.48724019 -1.30558569 -0.2723291 -0.83445805  1.015297761
## 5   -0.48724019 -1.30558569 -0.2723291 -0.83445805  1.227362043
## 6   -0.48724019 -1.30558569 -0.2723291 -0.83445805  0.206891639
## 7    0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.388026950
## 8    0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.160306916
## 9    0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.930285282
## 10   0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.399412952
## 11   0.04872402 -0.47618230 -0.2723291 -0.26489191  0.131459378
## 12   0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.392296701
## 13   0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.563086727
## 14  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.477691714
## 15  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 16  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.641365488
## 17  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.497617217
## 18  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.419338455
## 19  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.179354069
## 20  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.793653261
## 21  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.017103545
## 22  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.454919710
## 23  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.203004422
## 24  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.671253743
## 25  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.513272969
## 26  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.975829289
## 27  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.671253743
## 28  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.338213193
## 29  -0.48724019 -0.43682573 -0.2723291 -0.14407485  0.299402903
## 30  -0.48724019 -0.43682573 -0.2723291 -0.14407485  0.554164692
## 31  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.813578764
## 32  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.302631937
## 33  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.476268464
## 34  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.830657767
## 35  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 36  -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.500463717
## 37  -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.631402737
## 38  -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.618593485
## 39  -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.453496460
## 40   2.72854505 -1.19334657 -0.2723291 -1.09335175  0.441727925
## 41   2.72854505 -1.19334657 -0.2723291 -1.09335175  1.052302267
## 42  -0.48724019 -0.61611679 -0.2723291 -0.92075595  0.690796712
## 43  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.164576667
## 44  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.104800158
## 45  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.306901688
## 46  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.857699521
## 47  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.709681499
## 48  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.362408446
## 49  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -1.260479332
## 50  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.971559538
## 51   0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.457766211
## 52   0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.241432178
## 53   0.41317968 -0.80123846 -0.2723291 -0.99842406  0.322174907
## 54   0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.407952453
## 55   2.72854505 -1.04029322 -0.2723291 -1.24868797 -0.564509977
## 56   3.37170210 -1.44552019 -0.2723291 -1.30909650  1.372533565
## 57   3.15731641 -1.51548743 -0.2723291 -1.24868797  0.139998879
## 58   3.80047346 -1.43094368 -0.2723291 -1.24005818  0.756266222
## 59   0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.198734672
## 60   0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.509003218
## 61   0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.773727758
## 62   0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.453496460
## 63   0.58468822 -0.87557866 -0.2723291 -0.87760700  0.243896145
## 64   0.58468822 -0.87557866 -0.2723291 -0.87760700  0.679410711
## 65   0.26310970 -1.42219777 -0.2723291 -1.19604625  1.166162284
## 66   2.94293073 -1.13212523 -0.2723291 -1.35224545  0.007636609
## 67   2.94293073 -1.13212523 -0.2723291 -1.35224545 -0.708258248
## 68   0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.578742479
## 69   0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.982945540
## 70   0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.568779727
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## 295 -0.48724019  0.40569652 -0.2723291 -1.01568364 -0.392296701
## 296 -0.48724019  0.40569652 -0.2723291 -1.01568364  0.559857693
## 297 -0.48724019  0.40569652 -0.2723291 -1.01568364  0.376258415
## 298 -0.48724019  0.40569652 -0.2723291 -1.01568364 -0.703988498
## 299  2.51415937 -1.29683979 -0.2723291 -1.33498587  0.085915371
## 300  2.51415937 -1.29683979 -0.2723291 -1.33498587  1.076497520
## 301  2.51415937 -1.29683979 -0.2723291 -1.33498587  0.834544984
## 302  0.97058245 -0.73564417 -0.2723291 -1.05020280  0.434611674
## 303  0.97058245 -0.73564417 -0.2723291 -1.05020280  0.299402903
## 304  0.97058245 -0.73564417 -0.2723291 -1.05020280  0.992525758
## 305  0.92770532 -1.30558569 -0.2723291 -0.71364099  1.354031312
## 306  0.92770532 -1.30558569 -0.2723291 -0.71364099  0.471616179
## 307  0.92770532 -1.30558569 -0.2723291 -0.71364099  1.615909352
## 308  0.92770532 -1.30558569 -0.2723291 -0.71364099  0.803233479
## 309 -0.48724019 -0.18027916 -0.2723291 -0.09229612  0.498657933
## 310 -0.48724019 -0.18027916 -0.2723291 -0.09229612 -0.444956959
## 311 -0.48724019 -0.18027916 -0.2723291 -0.09229612 -1.866783923
## 312 -0.48724019 -0.18027916 -0.2723291 -0.09229612 -0.231469427
## 313 -0.48724019 -0.18027916 -0.2723291 -0.09229612 -0.372371198
## 314 -0.48724019 -0.18027916 -0.2723291 -0.09229612 -0.026521396
## 315 -0.48724019 -0.18027916 -0.2723291 -0.09229612  0.401876919
## 316 -0.48724019 -0.18027916 -0.2723291 -0.09229612 -0.824964766
## 317 -0.48724019 -0.18027916 -0.2723291 -0.09229612 -0.527505471
## 318 -0.48724019 -0.18027916 -0.2723291 -0.09229612 -0.715374500
## 319 -0.48724019 -0.18027916 -0.2723291 -0.09229612  0.138575629
## 320 -0.48724019 -0.18027916 -0.2723291 -0.09229612 -0.244278679
## 321 -0.48724019 -0.54760720 -0.2723291 -0.53241540  0.201198639
## 322 -0.48724019 -0.54760720 -0.2723291 -0.53241540  0.130036128
## 323 -0.48724019 -0.54760720 -0.2723291 -0.53241540 -0.346752694
## 324 -0.48724019 -0.54760720 -0.2723291 -0.53241540 -0.820695015
## 325 -0.48724019 -0.54760720 -0.2723291 -0.53241540  0.185542886
## 326 -0.48724019 -0.54760720 -0.2723291 -0.53241540  0.208314890
## 327 -0.48724019 -0.54760720 -0.2723291 -0.53241540  0.038948114
## 328 -0.48724019 -0.54760720 -0.2723291 -0.53241540 -0.286976185
## 329 -0.48724019 -1.15107469 -0.2723291 -0.81719847 -0.592974981
## 330 -0.48724019 -1.15107469 -0.2723291 -0.81719847  0.068836369
## 331 -0.48724019 -1.15107469 -0.2723291 -0.81719847 -0.200157922
## 332  1.01345959 -0.74001712 -0.2723291 -1.00791683 -0.823541516
## 333  1.01345959 -0.74001712 -0.2723291 -1.00791683 -0.360985196
## 334 -0.48724019 -0.86683276 -0.2723291 -0.34256002  0.044641115
## 335 -0.48724019 -0.86683276 -0.2723291 -0.34256002  0.036101614
## 336 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.352445695
## 337 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.591551731
## 338 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.554547225
## 339 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.321134190
## 340 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.426454706
## 341 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.450649960
## 342  1.01345959 -1.40179066 -0.2723291 -0.97253469  1.361147563
## 343 -0.48724019 -1.34785757 -0.2723291 -0.31667065  0.363449163
## 344  1.87100232 -1.07236154 -0.2723291 -0.61008351  0.585476196
## 345  1.87100232 -1.07236154 -0.2723291 -0.61008351  0.838814735
## 346 -0.48724019 -0.98344483 -0.2723291 -0.97253469 -0.385180450
## 347 -0.48724019 -0.98344483 -0.2723291 -0.97253469 -0.550277475
## 348  3.15731641 -1.01842846 -0.2723291 -1.08472196  0.329291158
## 349  2.94293073 -1.33036576 -0.2723291 -1.03294322  0.498657933
## 350  1.22784527 -1.44114723 -0.2723291 -1.08472196  0.931325998
## 351  1.22784527 -1.44114723 -0.2723291 -1.08472196  0.292286652
## 352  2.08538800 -1.37701059 -0.2723291 -1.24005818  0.418955921
## 353  2.08538800 -1.37701059 -0.2723291 -1.24005818 -0.570202978
## 354  3.37170210 -1.32890811 -0.2723291 -1.24868797  0.631020203
## 355  2.94293073 -1.34494227 -0.2723291 -1.22279860 -0.884741275
## 356  2.94293073 -1.34494227 -0.2723291 -1.22279860 -0.496193967
## 357 -0.48724019  1.01499462  3.6647712  1.85803641 -0.103376907
## 358 -0.48724019  1.01499462  3.6647712  1.85803641  0.157077882
## 359 -0.48724019  1.01499462  3.6647712  1.85803641 -0.224353176
## 360 -0.48724019  1.01499462 -0.2723291  1.85803641 -0.245701929
## 361 -0.48724019  1.01499462 -0.2723291  1.85803641  0.161347633
## 362 -0.48724019  1.01499462 -0.2723291  1.85803641 -0.047870149
## 363 -0.48724019  1.01499462 -0.2723291  1.85803641 -1.313139590
## 364 -0.48724019  1.01499462  3.6647712  1.85803641 -0.685486245
## 365 -0.48724019  1.01499462  3.6647712  1.40928733  3.551529643
## 366 -0.48724019  1.01499462 -0.2723291  1.40928733 -3.876413226
## 367 -0.48724019  1.01499462 -0.2723291  1.40928733 -1.881016425
## 368 -0.48724019  1.01499462 -0.2723291  0.65849561 -3.446591661
## 369 -0.48724019  1.01499462 -0.2723291  0.65849561 -1.871053674
## 370 -0.48724019  1.01499462  3.6647712  0.65849561  0.566973944
## 371 -0.48724019  1.01499462  3.6647712  0.65849561  1.040916265
## 372 -0.48724019  1.01499462 -0.2723291  0.65849561 -0.097683907
## 373 -0.48724019  1.01499462  3.6647712  0.97779784 -0.583012230
## 374 -0.48724019  1.01499462 -0.2723291  0.97779784 -1.962141687
## 375 -0.48724019  1.01499462 -0.2723291  0.97779784 -3.055197852
## 376 -0.48724019  1.01499462 -0.2723291  1.00368721  1.463621579
## 377 -0.48724019  1.01499462 -0.2723291  1.00368721  0.518583436
## 378 -0.48724019  1.01499462 -0.2723291  1.00368721  0.724954717
## 379 -0.48724019  1.01499462 -0.2723291  1.00368721  0.135729129
## 380 -0.48724019  1.01499462 -0.2723291  1.00368721 -0.087721155
## 381 -0.48724019  1.01499462 -0.2723291  1.00368721  0.972600255
## 382 -0.48724019  1.01499462 -0.2723291  1.00368721  0.370565414
## 383 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.065494052
## 384 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.088266056
## 385 -0.48724019  1.01499462 -0.2723291  1.25395112 -2.727850303
## 386 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.434115858
## 387 -0.48724019  1.01499462 -0.2723291  1.25395112 -2.323647242
## 388 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.828356167
## 389 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.999146193
## 390 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.273288584
## 391 -0.48724019  1.01499462 -0.2723291  1.25395112 -0.813578764
## 392 -0.48724019  1.01499462 -0.2723291  1.25395112 -0.332520192
## 393 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.777119159
## 394 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.130418661
## 395 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.565933227
## 396 -0.48724019  1.01499462 -0.2723291  1.19354259  0.265244898
## 397 -0.48724019  1.01499462 -0.2723291  1.19354259  0.171310384
## 398 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.765188257
## 399 -0.48724019  1.01499462 -0.2723291  1.19354259 -1.183623820
## 400 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.615746985
## 401 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.423608206
## 402 -0.48724019  1.01499462 -0.2723291  1.19354259  0.083068871
## 403 -0.48724019  1.01499462 -0.2723291  1.19354259  0.169887134
## 404 -0.48724019  1.01499462 -0.2723291  1.19354259 -1.331641842
## 405 -0.48724019  1.01499462 -0.2723291  1.19354259 -1.072610303
## 406 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.856276271
## 407 -0.48724019  1.01499462 -0.2723291  0.90012973 -3.055197852
## 408 -0.48724019  1.01499462 -0.2723291  0.90012973 -0.963020037
## 409 -0.48724019  1.01499462 -0.2723291  0.36508275 -0.950210785
## 410 -0.48724019  1.01499462 -0.2723291  0.36508275  0.807503230
## 411 -0.48724019  1.01499462 -0.2723291  0.36508275 -0.750955755
## 412 -0.48724019  1.01499462 -0.2723291  0.36508275  0.529969438
## 413 -0.48724019  1.01499462 -0.2723291  0.36508275 -2.357805247
## 414 -0.48724019  1.01499462 -0.2723291  0.36508275 -1.607752384
## 415 -0.48724019  1.01499462 -0.2723291  1.19354259 -2.512939520
## 416 -0.48724019  1.01499462 -0.2723291  1.07272553  0.212584640
## 417 -0.48724019  1.01499462 -0.2723291  1.07272553  0.707875715
## 418 -0.48724019  1.01499462 -0.2723291  1.07272553 -1.395688102
## 419 -0.48724019  1.01499462 -0.2723291  1.07272553 -0.466305712
## 420 -0.48724019  1.01499462 -0.2723291  1.40928733  0.767652224
## 421 -0.48724019  1.01499462 -0.2723291  1.40928733  0.179849885
## 422 -0.48724019  1.01499462 -0.2723291  1.40928733 -0.396566452
## 423 -0.48724019  1.01499462 -0.2723291  0.51178918 -0.906090028
## 424 -0.48724019  1.01499462 -0.2723291  0.51178918 -0.258511181
## 425 -0.48724019  1.01499462 -0.2723291  0.25289548 -1.024219796
## 426 -0.48724019  1.01499462 -0.2723291  1.07272553 -0.553123975
## 427 -0.48724019  1.01499462 -0.2723291  0.25289548 -0.637095738
## 428 -0.48724019  1.01499462 -0.2723291  1.07272553 -0.117609410
## 429 -0.48724019  1.01499462 -0.2723291  1.07272553 -0.130418661
## 430 -0.48724019  1.01499462 -0.2723291  1.07272553  0.135729129
## 431 -0.48724019  1.01499462 -0.2723291  0.25289548  0.090185122
## 432 -0.48724019  1.01499462 -0.2723291  0.25289548  0.780461476
## 433 -0.48724019  1.01499462 -0.2723291  0.25289548  0.199775388
## 434 -0.48724019  1.01499462 -0.2723291  1.36613839  0.215431141
## 435 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.109069908
## 436 -0.48724019  1.01499462 -0.2723291  1.59914271  0.490118432
## 437 -0.48724019  1.01499462 -0.2723291  1.59914271  0.251012396
## 438 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.188771920
## 439 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.497617217
## 440 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.935978283
## 441 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.664137492
## 442 -0.48724019  1.01499462 -0.2723291  1.59914271  0.172733634
## 443 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.093414156
## 444 -0.48724019  1.01499462 -0.2723291  1.59914271  0.285170401
## 445 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.612900484
## 446 -0.48724019  1.01499462 -0.2723291  1.59914271  0.248165896
## 447 -0.48724019  1.01499462 -0.2723291  1.59914271  0.080222370
## 448 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.047870149
## 449 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.141804663
## 450 -0.48724019  1.01499462 -0.2723291  1.36613839  0.188389387
## 451 -0.48724019  1.01499462 -0.2723291  1.36613839  0.660908458
## 452 -0.48724019  1.01499462 -0.2723291  1.36613839  0.527122938
## 453 -0.48724019  1.01499462 -0.2723291  1.36613839  0.017599361
## 454 -0.48724019  1.01499462 -0.2723291  1.36613839  1.577481596
## 455 -0.48724019  1.01499462 -0.2723291  1.36613839  0.631020203
## 456 -0.48724019  1.01499462 -0.2723291  1.36613839  0.342100410
## 457 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.439263958
## 458 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.496193967
## 459 -0.48724019  1.01499462 -0.2723291  1.36613839  0.023292362
## 460 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.289822685
## 461 -0.48724019  1.01499462 -0.2723291  1.36613839  0.592592447
## 462 -0.48724019  1.01499462 -0.2723291  1.36613839  0.130036128
## 463 -0.48724019  1.01499462 -0.2723291  1.36613839  0.046064365
## 464 -0.48724019  1.01499462 -0.2723291  1.36613839  0.325021407
## 465 -0.48724019  1.01499462 -0.2723291  0.86561057 -0.107646658
## 466 -0.48724019  1.01499462 -0.2723291  0.86561057 -0.748109254
## 467 -0.48724019  1.01499462 -0.2723291  0.86561057 -0.473421963
## 468 -0.48724019  1.01499462 -0.2723291  0.25289548 -0.400836202
## 469 -0.48724019  1.01499462 -0.2723291  0.21837632 -0.510426469
## 470 -0.48724019  1.01499462 -0.2723291  0.21837632 -0.813578764
## 471 -0.48724019  1.01499462 -0.2723291  0.21837632 -0.167423167
## 472 -0.48724019  1.01499462 -0.2723291 -0.19585359 -0.079181654
## 473 -0.48724019  1.01499462 -0.2723291  0.21837632  0.216854391
## 474 -0.48724019  1.01499462 -0.2723291  0.51178918  0.989679257
## 475 -0.48724019  1.01499462 -0.2723291  0.25289548 -1.220628326
## 476 -0.48724019  1.01499462 -0.2723291  0.25289548 -0.174539418
## 477 -0.48724019  1.01499462 -0.2723291  0.51178918  0.283747151
## 478 -0.48724019  1.01499462 -0.2723291  0.51178918 -1.395688102
## 479 -0.48724019  1.01499462 -0.2723291  0.51178918 -0.141804663
## 480 -0.48724019  1.01499462 -0.2723291  0.51178918 -0.079181654
## 481 -0.48724019  1.01499462 -0.2723291 -0.19585359 -0.060679401
## 482 -0.48724019  1.01499462 -0.2723291 -0.19585359  0.662331708
## 483 -0.48724019  1.01499462 -0.2723291 -0.19585359  1.104962525
## 484 -0.48724019  1.01499462 -0.2723291 -0.19585359 -0.743839504
## 485 -0.48724019  1.01499462 -0.2723291  0.24426569 -0.588705230
## 486 -0.48724019  1.01499462 -0.2723291  0.24426569  0.038948114
## 487 -0.48724019  1.01499462 -0.2723291  0.24426569 -0.242855428
## 488 -0.48724019  1.01499462 -0.2723291  0.24426569 -0.540314723
## 489 -0.48724019  2.42017014 -0.2723291  0.46864023 -1.182200570
## 490 -0.48724019  2.42017014 -0.2723291  0.46864023 -1.239130578
## 491 -0.48724019  2.42017014 -0.2723291  0.46864023 -1.695993897
## 492 -0.48724019  2.42017014 -0.2723291  0.46864023 -0.429301206
## 493 -0.48724019  2.42017014 -0.2723291  0.46864023 -0.429301206
## 494 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.822118266
## 495 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.510426469
## 496 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.874778524
## 497 -0.48724019 -0.21088983 -0.2723291  0.26152527 -1.273288584
## 498 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.698295497
## 499 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.378064199
## 500 -0.48724019 -0.21088983 -0.2723291  0.26152527 -1.018526795
## 501 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.366678197
## 502 -0.48724019  0.11562398 -0.2723291  0.15796779  0.438881424
## 503 -0.48724019  0.11562398 -0.2723291  0.15796779 -0.234315927
## 504 -0.48724019  0.11562398 -0.2723291  0.15796779  0.983986256
## 505 -0.48724019  0.11562398 -0.2723291  0.15796779  0.724954717
## 506 -0.48724019  0.11562398 -0.2723291  0.15796779 -0.362408446
##              age           dis        rad         tax     ptratio
## 1   -0.119894767  0.1400749840 -0.9818712 -0.66594918 -1.45755797
## 2    0.366803426  0.5566090496 -0.8670245 -0.98635338 -0.30279450
## 3   -0.265548971  0.5566090496 -0.8670245 -0.98635338 -0.30279450
## 4   -0.809087830  1.0766711351 -0.7521778 -1.10502160  0.11292035
## 5   -0.510674339  1.0766711351 -0.7521778 -1.10502160  0.11292035
## 6   -0.350809969  1.0766711351 -0.7521778 -1.10502160  0.11292035
## 7   -0.070159185  0.8384142195 -0.5224844 -0.57694801 -1.50374851
## 8    0.977840575  1.0236248974 -0.5224844 -0.57694801 -1.50374851
## 9    1.116389695  1.0861216287 -0.5224844 -0.57694801 -1.50374851
## 10   0.615481336  1.3283202075 -0.5224844 -0.57694801 -1.50374851
## 11   0.913894827  1.2117799501 -0.5224844 -0.57694801 -1.50374851
## 12   0.508905089  1.1547920492 -0.5224844 -0.57694801 -1.50374851
## 13  -1.050660656  0.7863652700 -0.5224844 -0.57694801 -1.50374851
## 14  -0.240681180  0.4333252240 -0.6373311 -0.60068166  1.17530274
## 15   0.565745754  0.3166899868 -0.6373311 -0.60068166  1.17530274
## 16  -0.428965883  0.3341187865 -0.6373311 -0.60068166  1.17530274
## 17  -1.395257187  0.3341187865 -0.6373311 -0.60068166  1.17530274
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## 451  0.853501620 -0.6987869171  1.6596029  1.52941294  0.80577843
## 452  1.052443947 -0.6837801032  1.6596029  1.52941294  0.80577843
## 453  0.825081288 -0.6776064140  1.6596029  1.52941294  0.80577843
## 454  1.091521905 -0.6374774338  1.6596029  1.52941294  0.80577843
## 455  0.906789743 -0.6168668096  1.6596029  1.52941294  0.80577843
## 456  0.636796585 -0.6455032298  1.6596029  1.52941294  0.80577843
## 457  0.686532167 -0.5767378294  1.6596029  1.52941294  0.80577843
## 458  0.416539008 -0.4824228534  1.6596029  1.52941294  0.80577843
## 459  0.537325421 -0.4805707466  1.6596029  1.52941294  0.80577843
## 460  0.562193212 -0.5117241325  1.6596029  1.52941294  0.80577843
## 461  0.761135540 -0.5687120333  1.6596029  1.52941294  0.80577843
## 462  0.704294875 -0.5831489682  1.6596029  1.52941294  0.80577843
## 463  0.512457630 -0.5036983364  1.6596029  1.52941294  0.80577843
## 464  0.757582998 -0.4717851119  1.6596029  1.52941294  0.80577843
## 465 -0.112789684 -0.3949464255  1.6596029  1.52941294  0.80577843
## 466 -0.723826832 -0.3459843207  1.6596029  1.52941294  0.80577843
## 467  0.572850837 -0.4385896596  1.6596029  1.52941294  0.80577843
## 468  0.920999910 -0.5958762661  1.6596029  1.52941294  0.80577843
## 469  0.086152643 -0.4210658801  1.6596029  1.52941294  0.80577843
## 470 -0.421860800 -0.4612898402  1.6596029  1.52941294  0.80577843
## 471  0.547983046 -0.3617034834  1.6596029  1.52941294  0.80577843
## 472  0.786003330 -0.3304076278  1.6596029  1.52941294  0.80577843
## 473  0.228254306 -0.4267171803  1.6596029  1.52941294  0.80577843
## 474 -0.034633770 -0.5993905200  1.6596029  1.52941294  0.80577843
## 475  0.952972784 -0.6483526248  1.6596029  1.52941294  0.80577843
## 476  1.024023615 -0.7546350600  1.6596029  1.52941294  0.80577843
## 477  0.889027036 -0.7074775720  1.6596029  1.52941294  0.80577843
## 478  1.020471073 -0.8046419430  1.6596029  1.52941294  0.80577843
## 479  0.999155824 -0.7714939807  1.6596029  1.52941294  0.80577843
## 480  0.690084708 -0.8756393696  1.6596029  1.52941294  0.80577843
## 481 -0.137657475 -0.1761128861  1.6596029  1.52941294  0.80577843
## 482  0.224701764 -0.2200410597  1.6596029  1.52941294  0.80577843
## 483  0.299305137 -0.1825715149  1.6596029  1.52941294  0.80577843
## 484 -1.004477616  0.1440166471  1.6596029  1.52941294  0.80577843
## 485 -0.947636951 -0.0337381137  1.6596029  1.52941294  0.80577843
## 486 -0.592382795  0.0933923952  1.6596029  1.52941294  0.80577843
## 487  0.398776300 -0.1183176566  1.6596029  1.52941294  0.80577843
## 488 -0.546199754 -0.3052379716  1.6596029  1.52941294  0.80577843
## 489  0.857054162 -0.9375187319 -0.6373311  1.79641644  0.75958789
## 490  1.055996489 -0.9686246278 -0.6373311  1.79641644  0.75958789
## 491  1.045338864 -0.9367114033 -0.6373311  1.79641644  0.75958789
## 492  1.073759197 -0.9151034909 -0.6373311  1.79641644  0.75958789
## 493  0.530220338 -0.8002728706 -0.6373311  1.79641644  0.75958789
## 494 -0.517779422 -0.6711952751 -0.4076377 -0.10227512  0.34387304
## 495 -0.922769160 -0.6711952751 -0.4076377 -0.10227512  0.34387304
## 496 -1.413019895 -0.4732098094 -0.4076377 -0.10227512  0.34387304
## 497  0.153650933 -0.4732098094 -0.4076377 -0.10227512  0.34387304
## 498  0.071942477 -0.4285217972 -0.4076377 -0.10227512  0.34387304
## 499 -0.116342226 -0.6581830377 -0.4076377 -0.10227512  0.34387304
## 500  0.174966182 -0.6625521101 -0.4076377 -0.10227512  0.34387304
## 501  0.395223759 -0.6158695213 -0.4076377 -0.10227512  0.34387304
## 502  0.018654354 -0.6251775451 -0.9818712 -0.80241764  1.17530274
## 503  0.288647512 -0.7159307773 -0.9818712 -0.80241764  1.17530274
## 504  0.796660955 -0.7729186782 -0.9818712 -0.80241764  1.17530274
## 505  0.736267749 -0.6677760011 -0.9818712 -0.80241764  1.17530274
## 506  0.434301716 -0.6126402069 -0.9818712 -0.80241764  1.17530274
##            black        lstat         medv
## 1    0.440615895 -1.074498970  0.159527789
## 2    0.440615895 -0.491952525 -0.101423917
## 3    0.396035074 -1.207532413  1.322937477
## 4    0.415751408 -1.360170785  1.181588636
## 5    0.440615895 -1.025486649  1.486032293
## 6    0.410165113 -1.042290874  0.670558212
## 7    0.426376321 -0.031236706  0.039924923
## 8    0.440615895  0.909799859  0.496590409
## 9    0.328123258  2.419379350 -0.655946292
## 10   0.328999540  0.622727693 -0.394994586
## 11   0.392639483  1.091845624 -0.819041108
## 12   0.440615895  0.086392864 -0.394994586
## 13   0.370513376  0.428078760 -0.090550930
## 14   0.440615895 -0.615183504 -0.231899770
## 15   0.255720500 -0.335113097 -0.471105500
## 16   0.426595391 -0.585776111 -0.286264709
## 17   0.330533033 -0.850442645  0.061670899
## 18   0.329437681  0.282442149 -0.547216415
## 19  -0.741378303 -0.134862757 -0.253645746
## 20   0.375442459 -0.192277190 -0.471105500
## 21   0.217930861  1.171665689 -0.971262937
## 22   0.392749018  0.164812578 -0.318883672
## 23   0.440615895  0.849584722 -0.797295133
## 24   0.414765591  1.012025558 -0.873406047
## 25   0.412465352  0.510699530 -0.753803182
## 26  -0.583319029  0.540106923 -0.938643973
## 27   0.221326452  0.302047077 -0.645073304
## 28  -0.550896614  0.647934029 -0.840787084
## 29   0.342472368  0.020576319 -0.449359525
## 30   0.258020739 -0.094252548 -0.166661844
## 31   0.038293155  1.392921311 -1.069119826
## 32   0.219683424  0.054184768 -0.873406047
## 33  -1.359047220  2.108501199 -1.014754888
## 34   0.022958229  0.797771697 -1.025627875
## 35  -1.186967442  1.076441751 -0.982135924
## 36   0.440615895 -0.416333515 -0.394994586
## 37   0.228774844 -0.174072614 -0.275391721
## 38   0.440615895 -0.543765550 -0.166661844
## 39   0.402607185 -0.353317674  0.235638703
## 40   0.426704926 -1.166922204  0.898890955
## 41   0.426595391 -1.494604580  1.344683452
## 42   0.314759966 -1.094103899  0.442225470
## 43   0.292414788 -0.958269752  0.300876629
## 44   0.413889309 -0.730012370  0.235638703
## 45   0.358354970 -0.434538092 -0.144915868
## 46   0.440615895 -0.342114857 -0.351502635
## 47   0.440615895  0.209623843 -0.275391721
## 48   0.395049257  0.860787538 -0.645073304
## 49   0.440615895  2.542610329 -0.884279035
## 50   0.440615895  0.496696010 -0.340629648
## 51   0.425938180  0.111599201 -0.308010684
## 52   0.408522085 -0.451342316 -0.221026782
## 53   0.440615895 -1.032488409  0.268257666
## 54   0.440615895 -0.591377519  0.094289862
## 55   0.440615895  0.300646725 -0.394994586
## 56   0.429990982 -1.098304955  1.399048391
## 57   0.440615895 -0.963871160  0.235638703
## 58   0.396801820 -1.218735230  0.985874857
## 59   0.372485009 -0.811232788  0.083416874
## 60   0.440615895 -0.480749709 -0.318883672
## 61   0.421009097  0.069588640 -0.416740562
## 62   0.234470674  0.250234052 -0.710311231
## 63   0.440615895 -0.829437365 -0.036185991
## 64   0.426157250 -0.441539852  0.268257666
## 65   0.400526017 -0.644590896  1.138096685
## 66   0.440615895 -1.117909883  0.105162850
## 67   0.440615895 -0.337913801 -0.340629648
## 68   0.433057967 -0.637589136 -0.057931966
## 69   0.440615895  0.061186528 -0.558089402
## 70   0.440615895 -0.540964846 -0.177534831
## 71   0.296358054 -0.830837717  0.181273764
## 72   0.221983663 -0.388326475 -0.090550930
## 73   0.375004318 -0.998879961  0.029051936
## 74   0.224502972 -0.716008850  0.094289862
## 75   0.418927928 -0.822435605  0.170400776
## 76   0.290881295 -0.519959566 -0.123169893
## 77   0.186056121 -0.095652900 -0.275391721
## 78   0.331737920 -0.333712745 -0.188407819
## 79   0.325603949 -0.043839875 -0.144915868
## 80   0.431414939 -0.497553933 -0.242772758
## 81   0.440615895 -1.031088057  0.594447298
## 82   0.426704926 -0.760820115  0.148654801
## 83   0.440615895 -0.830837717  0.246511690
## 84   0.372046868 -0.720209906  0.039924923
## 85   0.440615895 -0.424735627  0.148654801
## 86   0.390229709 -0.857444405  0.442225470
## 87   0.430648193  0.028978431 -0.003567028
## 88   0.421447237 -0.589977167 -0.036185991
## 89   0.440615895 -1.001680665  0.116035838
## 90   0.431414939 -0.973673624  0.670558212
## 91   0.388915287 -0.538164142  0.007305960
## 92   0.403921608 -0.623585616 -0.057931966
## 93   0.419913745 -0.629187024  0.039924923
## 94   0.434372389 -0.902255670  0.268257666
## 95   0.440615895 -0.288901480 -0.210153795
## 96   0.014304949 -0.840640181  0.637939249
## 97   0.385081555 -0.183875078 -0.123169893
## 98   0.440615895 -1.182326077  1.757856987
## 99   0.403702537 -1.271948607  2.312379361
## 100  0.440615895 -0.905056374  1.159842661
## 101  0.417175365 -0.452742668  0.540082359
## 102  0.426157250 -0.697804274  0.431352482
## 103 -3.131326538 -0.283300072 -0.427613550
## 104  0.413998845  0.110198849 -0.351502635
## 105  0.394501581 -0.045240227 -0.264518733
## 106  0.409398367  0.534505515 -0.329756660
## 107  0.427143067  0.841182610 -0.329756660
## 108  0.339733988  0.201221731 -0.231899770
## 109  0.422433054 -0.053642339 -0.297137697
## 110  0.378509444  0.405673128 -0.340629648
## 111  0.403264396  0.048583360 -0.090550930
## 112  0.426266786 -0.349116618  0.029051936
## 113  0.419256534  0.498096362 -0.405867574
## 114  0.440615895  0.621327341 -0.416740562
## 115  0.351235183 -0.308506409 -0.438486537
## 116 -0.128857540  0.435080520 -0.460232513
## 117  0.401183228 -0.085850436 -0.144915868
## 118  0.414436985 -0.329511689 -0.362375623
## 119 -0.197645637  0.380466791 -0.231899770
## 120  0.381466894  0.134004834 -0.351502635
## 121  0.355726125  0.240431588 -0.057931966
## 122  0.229979731  0.226428068 -0.242772758
## 123  0.234580209  0.738956911 -0.221026782
## 124  0.149361834  1.786420232 -0.568962390
## 125  0.248710248  0.689944590 -0.405867574
## 126  0.310488093  0.302047077 -0.123169893
## 127  0.028654058  2.045485358 -0.742930194
## 128  0.388148541  0.635330861 -0.688565255
## 129  0.440615895  0.383267495 -0.492851476
## 130  0.440615895  0.796371345 -0.895152022
## 131  0.420242350 -0.007430722 -0.362375623
## 132  0.440615895 -0.055042691 -0.318883672
## 133  0.318593697 -0.214682823  0.050797911
## 134  0.350687507  0.332854822 -0.449359525
## 135 -1.028689097  0.652135085 -0.753803182
## 136  0.416189548  0.603122764 -0.481978488
## 137  0.236332772  0.594720652 -0.558089402
## 138  0.409726972  0.271239333 -0.590708366
## 139  0.387381794  1.213676250 -1.003881900
## 140  0.440615895  0.813175569 -0.514597451
## 141  0.344005860  1.611376228 -0.927770986
## 142  0.440615895  3.046737061 -0.884279035
## 143  0.440615895  1.983869868 -0.993008912
## 144  0.440615895  1.927855787 -0.753803182
## 145  0.440615895  2.329756820 -1.166976716
## 146 -2.012862748  2.121104367 -0.949516961
## 147 -2.052733556  0.559711851 -0.753803182
## 148  0.383767133  2.363365269 -0.862533059
## 149  0.003460966  2.193922673 -0.514597451
## 150 -0.052840120  1.231880827 -0.775549157
## 151  0.176636095  0.202622083 -0.112296905
## 152 -0.165113687  0.087793216 -0.318883672
## 153 -0.146711775 -0.074647619 -0.786422145
## 154 -1.037561447  0.439281577 -0.340629648
## 155 -0.390537100  0.345457990 -0.601581353
## 156 -2.942816482  0.331454470 -0.753803182
## 157 -2.936025300  0.488293898 -1.025627875
## 158  0.074001626 -1.129112700  2.040554668
## 159 -0.030494942 -0.871447926  0.192146752
## 160  0.083640722 -0.737014131  0.083416874
## 161 -0.194469117 -1.001680665  0.485717421
## 162  0.194490331 -1.529613381  2.986504601
## 163  0.360764744 -1.503006692  2.986504601
## 164  0.348058662 -1.306957408  2.986504601
## 165  0.421009097 -0.141864517  0.018178948
## 166 -1.276238619 -0.398128939  0.268257666
## 167  0.138298780 -1.253744030  2.986504601
## 168 -1.413705278 -0.071846915  0.137781813
## 169 -0.652654802 -0.217483527  0.137781813
## 170 -0.291736362 -0.186675782 -0.025313003
## 171 -0.705231691  0.248833700 -0.558089402
## 172 -0.093587210 -0.087250788 -0.373248611
## 173  0.440615895  0.285242853  0.061670899
## 174  0.425280969 -0.505956045  0.116035838
## 175  0.400416482 -0.421934923  0.007305960
## 176  0.375551994 -1.025486649  0.746669127
## 177  0.400416482 -0.356118378  0.072543887
## 178  0.426376321 -0.891052854  0.224765715
## 179  0.378947585 -0.802830676  0.801034065
## 180  0.440615895 -1.066096858  1.594762170
## 181  0.425938180 -0.713208146  1.877459852
## 182  0.440615895 -0.448541612  1.486032293
## 183  0.410165113 -1.096904603  1.670873085
## 184  0.440615895 -0.976474328  1.083731747
## 185  0.375990135  0.185817859  0.420479494
## 186  0.333380947  0.069588640  0.768415102
## 187  0.393844370 -1.148717628  2.986504601
## 188  0.407426733 -0.836439125  1.029366808
## 189  0.286609423 -1.133313756  0.790161078
## 190  0.440615895 -1.017084537  1.344683452
## 191  0.230089266 -1.057694746  1.573016195
## 192  0.361860096 -1.115109179  0.866271992
## 193  0.370403840 -1.369973249  1.507778268
## 194  0.401949974 -1.067497210  0.931509918
## 195  0.219354818 -1.158520092  0.714050163
## 196  0.411370000 -1.355969729  2.986504601
## 197  0.440615895 -1.200530653  1.170715648
## 198 -0.025894464 -0.566171183  0.844526016
## 199  0.389134357 -0.844841237  1.312064489
## 200  0.440615895 -1.133313756  1.344683452
## 201  0.302601560 -1.148717628  1.127223698
## 202  0.406331382 -0.731412722  0.170400776
## 203  0.423966547 -1.336364800  2.149284545
## 204  0.395487398 -1.238340158  2.823409785
## 205  0.371061052 -1.368572897  2.986504601
## 206  0.440615895 -0.249691623  0.007305960
## 207  0.418380252 -0.235688103  0.203019739
## 208  0.358793111  0.757161488 -0.003567028
## 209  0.269960074  0.281041797  0.203019739
## 210  0.440615895  1.461538560 -0.275391721
## 211  0.400635552  0.646533677 -0.090550930
## 212  0.422433054  1.586169891 -0.351502635
## 213  0.375332924  0.472890025 -0.014440015
## 214  0.319141373 -0.458344076  0.605320286
## 215 -0.084824395  2.366165973  0.126908825
## 216  0.404797889 -0.445740908  0.268257666
## 217  0.395706469  0.120001313  0.083416874
## 218  0.395487398 -0.414933163  0.670558212
## 219  0.440615895  0.737556559 -0.112296905
## 220  0.406002776 -0.301504649  0.050797911
## 221  0.383657598 -0.412132459  0.453098458
## 222  0.422433054  1.233281179 -0.090550930
## 223  0.369308489 -0.381324714  0.540082359
## 224  0.440615895 -0.707606738  0.822780041
## 225  0.310816699 -1.192128541  2.421109239
## 226  0.277408467 -1.123511291  2.986504601
## 227  0.336338397 -1.333564096  1.638254121
## 228  0.168749562 -0.881250390  0.985874857
## 229  0.228227168 -1.222936286  2.627696006
## 230  0.259225626 -1.245341918  0.975001869
## 231  0.237428124 -0.140464165  0.192146752
## 232  0.213220848 -1.036689465  0.996747845
## 233  0.320236725 -1.425987330  2.084046619
## 234  0.244000235 -1.218735230  2.801663810
## 235  0.038621760 -0.644590896  0.703177176
## 236  0.219902494 -0.248291271  0.159527789
## 237  0.348058662 -0.435938444  0.279130654
## 238  0.365803363 -1.109507771  0.975001869
## 239  0.249038854 -0.881250390  0.126908825
## 240  0.296905730 -0.739814835  0.083416874
## 241  0.378728515 -0.178273670 -0.057931966
## 242  0.415641872 -0.035437762 -0.264518733
## 243  0.176088420 -0.200679302 -0.036185991
## 244  0.197557316 -1.045091578  0.126908825
## 245  0.173240505 -0.021434242 -0.536343427
## 246  0.355507055  0.813175569 -0.438486537
## 247  0.367008250 -0.489151821  0.192146752
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boston_scaled<-data.frame(boston_scaled, crime)

Divide the dataset into train and test

n<-nrow(boston_scaled)
ind<-sample(n, size = n*0.8)
train<-boston_scaled[ind, ]
test<-boston_scaled[-ind, ]
correct_classes<-test$crime

Linear Discriminant Analysis

lda.fit<-lda(crime~., data=train)
lda.fit
## Call:
## lda(crime ~ ., data = train)
## 
## Prior probabilities of groups:
##       low   med_low  med_high      high 
## 0.2450495 0.2747525 0.2351485 0.2450495 
## 
## Group means:
##                   zn      indus        chas        nox         rm
## low       0.87075235 -0.8612966 -0.07348562 -0.8542804  0.4127885
## med_low  -0.08821242 -0.2923476  0.01142590 -0.5625108 -0.1301879
## med_high -0.35906012  0.1541520  0.18354570  0.3596324  0.1037884
## high     -0.48724019  1.0171737 -0.03371693  1.1004455 -0.4392496
##                 age        dis        rad        tax     ptratio
## low      -0.8595770  0.8624144 -0.6918543 -0.7471590 -0.38817701
## med_low  -0.3703329  0.3654480 -0.5473161 -0.4770957 -0.07100954
## med_high  0.4007956 -0.3845586 -0.4209357 -0.3293062 -0.30619801
## high      0.8024383 -0.8284699  1.6375616  1.5136504  0.78011702
##               black       lstat        medv
## low       0.3812257 -0.74900907  0.49691989
## med_low   0.3221067 -0.13949275  0.02395828
## med_high  0.1037514  0.07056152  0.15952779
## high     -0.7418906  0.87211766 -0.69713185
## 
## Coefficients of linear discriminants:
##                 LD1           LD2        LD3
## zn       0.07430630  0.6739413323 -1.0503003
## indus    0.06577009 -0.1655276720  0.1665862
## chas    -0.09327788 -0.0164629386  0.1402870
## nox      0.39369334 -0.8011409088 -1.3236015
## rm      -0.06386765 -0.1076562202 -0.1896987
## age      0.22634748 -0.3139933045 -0.1162417
## dis     -0.02506278 -0.1964260098  0.1669998
## rad      3.38034484  0.9291448260 -0.1167412
## tax      0.03411753 -0.0184446646  0.6865095
## ptratio  0.07358559  0.0765276095 -0.2617097
## black   -0.08404990  0.0002793718  0.1145223
## lstat    0.23707185 -0.2793616534  0.3207011
## medv     0.19738866 -0.4312337227 -0.1689729
## 
## Proportion of trace:
##    LD1    LD2    LD3 
## 0.9576 0.0309 0.0115
lda.arrows<-function(x, myscale=1, arrow_heads= 0.1, color="red", tex=0.75, choices =c (1,2)){
  heads<-coef(x)
  arrows(x0 = 0, y0=0, x1 = myscale*heads[,choices[1]],y1=myscale*heads[,choices[2]], col=color, length= arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}

Convert target classes to numeric data

classes<-as.numeric(train$crime)

Plot the lda results

plot(lda.fit, dimen=2, col=classes, pch=classes)
lda.arrows(lda.fit, myscale=1)

Predict the classes with the LDA model and Cross-tabulate the results

lda.pred<-predict(lda.fit, newdata = test)
table(correct=correct_classes, predict=lda.pred$class)
##           predict
## correct    low med_low med_high high
##   low       17      11        0    0
##   med_low    1      13        1    0
##   med_high   0      13       16    2
##   high       0       0        0   28

Distance measures

library(MASS)
data("Boston")
boston_scaled<-scale(Boston)
boston_scaled<-as.data.frame(boston_scaled)
Euclidean distane matrix
dist_eu<-dist(boston_scaled)
summary(dist_eu)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.1343  3.4625  4.8241  4.9111  6.1863 14.3970

K-means

km<-kmeans(boston_scaled, centers=3)
pairs(boston_scaled, col=km$cluster)

Investigate the optimal number of clusters

set.seed(11)
k_max<-20
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled, k)$tot.withinss})
qqplot(x=1:k_max, y = twcss, geom="Line")
## Warning in plot.window(...): "geom" is not a graphical parameter
## Warning in plot.xy(xy, type, ...): "geom" is not a graphical parameter
## Warning in axis(side = side, at = at, labels = labels, ...): "geom" is not
## a graphical parameter

## Warning in axis(side = side, at = at, labels = labels, ...): "geom" is not
## a graphical parameter
## Warning in box(...): "geom" is not a graphical parameter
## Warning in title(...): "geom" is not a graphical parameter

Perform k-means with different number of clusters

km<-kmeans(boston_scaled, centers=6)
pairs(boston_scaled, col = km$cluster)

library(GGally)
## Registered S3 method overwritten by 'GGally':
##   method from   
##   +.gg   ggplot2
## 
## Attaching package: 'GGally'
## The following object is masked from 'package:dplyr':
## 
##     nasa
ggpairs(boston_scaled)

Super Bonus

model_predictors<- dplyr::select(train, -crime)
Check th dimensions
dim(model_predictors)
## [1] 404  13
dim(lda.fit$scaling)
## [1] 13  3
Matrix multiplication
matrix_product <- as.matrix(model_predictors)%*%lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
Create a 3D plot
library(plotly)
## 
## Attaching package: 'plotly'
## The following object is masked from 'package:ggplot2':
## 
##     last_plot
## The following object is masked from 'package:MASS':
## 
##     select
## The following object is masked from 'package:stats':
## 
##     filter
## The following object is masked from 'package:graphics':
## 
##     layout
plot_ly(x=matrix_product$LD1, y=matrix_product$LD2, z=matrix_product$LD3, type="scatter3d", mode="markers")
plot_ly(x=matrix_product$LD1, y=matrix_product$LD2, z=matrix_product$LD3, type="scatter3d", mode="markers", color = train$crime)
km <-kmeans(boston_scaled, centers = 6)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = km$cluster[ind])